 Solution

No. Let ${\lambda }_1>0$ be the First Dirichlet eigenvalue of $-\Delta$ on $B$, choose a corresponding nonzero eigenfunction $w$, and takeThen $L{u}_k=0$ and every $u_k$ vanishes on $\partial B$, so the boundary values converge uniformly to those of $v=0$. But $|u_k{|}_{0;B}\to \infty$, and hence no subsequence converges uniformly on $B$.